Torsional buckling and post buckling behavior of eccentrically stiffened functionally graded toroidal shell segments surrounded by an elastic medium

O R I G I NA L PA P E RDinh Gia Ninh · Dao Huy Bich · Bui Huy Kien Torsional buckling and post-buckling behavior of eccentrically stiffened functionally graded toroidal shell segments su

O R I G I NA L PA P E R

Dinh Gia Ninh · Dao Huy Bich · Bui Huy Kien

Torsional buckling and post-buckling behavior

of eccentrically stiffened functionally graded toroidal shell segments surrounded by an elastic medium

Received: 4 March 2015 / Revised: 11 May 2015

© Springer-Verlag Wien 2015

Abstract The nonlinear buckling and post-buckling problems of functionally graded stiffened toroidal shell

segments surrounded by an elastic medium under torsion based on an analytical approach are investigated The rings and stringers are attached to the shell, and material properties of the shell are assumed to be continuously graded in the thickness direction The classical shell theory with the geometrical nonlinearity in von Kármán sense and the smeared stiffeners technique are applied to establish theoretical formulations The three-term approximate solution of deflection is chosen more correctly, and the explicit expression to find critical load and post-buckling torsional load-deflection curves is given The effects of geometrical parameters and the effectiveness of stiffeners on the stability of the shell are investigated

1 Introduction

Functionally graded materials (FGMs) were known by Japanese scientists in 1984 [1] This composite material

is a mixture of ceramic and metallic constituent materials by continuously changing the volume fractions of their components The advantage of FGMs is that they are better than the traditional fiber-reinforced and laminated composite materials in avoiding the stress concentration FGMs are applied to heat-resistant, lightweight structures in aerospace, mechanical, and medical industries, etc Therefore, the buckling and vibration problems

of FGM structures have attracted much attention of researchers

On the research of the torsional problem, Sofiyev et al [2,3] pointed out the torsional vibration and buckling analysis of a cylindrical shell surrounded by an elastic medium The torsion of a circular cylindrical bar made

of either an isotropic compressible or an isotropic incompressible linear elastic material with material moduli varying only in the axial direction was taken into account by Batra [4] The torsional post-buckling analysis of FGM cylindrical shells in thermal environment based on a higher-order shear deformation theory with a von Kármán–Donell type of kinematic nonlinearity was given by Shen [5] Sofiyev and Schnack [6] presented the stability of a functionally graded cylindrical shell subjected to torsional loading varying as a linear function of time The modified Donnell-type dynamic stability and compatibility equations were applied The nonlinear buckling problem of FGM cylindrical shells under torsion load based on the nonlinear large deflection theory by using the energy method and the nonlinear strain–displacement relations of large deformation was studied by

D G Ninh (B)

School of Mechanical Engineering, Hanoi University of Science and Technology, Hanoi, Vietnam

E-mail: ninhdinhgia@gmail.com; ninh.dinhgia@hust.edu.vn

Tel.: +84 988 287 789

D H Bich

Vietnam National University, Hanoi, Vietnam

B H Kien

Faculty of Mechanical Engineering, Hanoi University of Industry, Hanoi, Vietnam

Huang and Han [7] Wang et al [8] carried out the exact solutions and transient behavior for torsional vibration

of functionally graded finite hollow cylinders The torsional analysis of functionally graded hollow tubes of arbitrary shape based on governing equations in terms of Prandtl’s stress function was investigated by Arghavan and Hematiyan [9] Tan [10] developed the torsional buckling loads of thin and thick shells of revolution based

on the classical thin shell theory and the first-order shear deformation shell theory The nonlinear buckling and post-buckling problems of functionally graded stiffened thin circular cylindrical shells only subjected to torsional load by the analytical approach based on the classical shell theory with the geometrical nonlinearity

in von Kármán sense were studied by Dung and Hoa [11] The torsional stability analysis for thin cylindrical shells with the functionally graded middle layer resting on a Winkler elastic foundation was given by Sofiyev and Adiguzel [12] The fundamental relations and basic equation of three-layered cylindrical shells with a

FG middle layer resting on a Winkler elastic foundation under torsional load were derived Zhang and Fu [13] addressed the torsional buckling characteristic of an elastic cylinder with a hard surface coating layer

by Navier’s equation and thin shell model Recently, Dung and Hoa [14] investigated the nonlinear buckling and post-buckling of functionally graded stiffened thin circular cylindrical shells surrounded by an elastic foundation in thermal environments under torsional load by an analytical approach

The nonlinear buckling and post-buckling of heat functionally graded cylindrical shells under combined axial compression and radial pressure were studied by Huang and Han [15] Bich et al [16] investigated the linear buckling of truncated conical panels made of functionally graded materials and subjected to axial compression, external pressure, and the combination of these loads The nonlinear buckling behavior of trun-cated conical shells made of FGM using the large deformation theory with the von Kármán–Donnell type

of kinematic nonlinearity subjected to a uniform axial compressive load was investigated by Sofiyev [17] Furthermore, Duc et al [18,19] presented an analytical approach to present the nonlinear static buckling and buckling for imperfect eccentrically stiffened FGM of shell structures on elastic foundations The post-buckling analysis of axially loaded functionally graded cylindrical shells in thermal environments using the classical shell theory with von Kármán–Donnell type of kinematic nonlinearity was pointed out by Shen [20] The dynamic buckling of imperfect FGM cylindrical shells with integrated surface-bonded sensor and actuator layers subjected to some complex combinations of thermo-electro-mechanical loads based on the general form

of Green’s strain tensor in curvilinear coordinates and a high-order shell theory proposed earlier was studied by Shariyat [21] Liew et al [22] calculated the post-buckling of FGM cylindrical shells under axial compression and thermal loads using the element-free kp-Ritz method Kernel shape functions were used to approximate field variables and formulations based on the Ritz procedure which leads to a system of nonlinear discrete equations and overcomes the shortcomings of the conventional Rayleigh–Ritz method, in which it is difficult

to choose appropriate global trial functions for problems with complicated boundary conditions The linear thermal buckling and free vibration for functionally graded cylindrical shells subjected to a clamped–clamped boundary condition with temperature-dependent material properties were investigated by Kadoli and Ganesan [23] The buckling behavior of FGM cylindrical shells subjected to pure bending load were taken into account

by Huang et al [24] Sofiyev et al [25] discussed the buckling of FGM hybrid truncated conical shells subjected

to hydrostatic pressure The author chose the available solution to satisfy the boundary condition, inserted them into the governing equations, and then used Galerkin’s method to lead to pairs of time-dependent differential equations Moreover, the thermal buckling of FGM sandwich plates was studied by Zenkour and Sobhy [26] using the sinusoidal shear deformation

The shell on an elastic foundation has been studied by many authors The simplest model for the elastic foundation is Winkler’s model [27] like a series of separated springs without coupling effects between each other, and then a shear layer to one-parameter model is added by a Pasternak [28] Bagherizadeh et al [29] investigated the mechanical buckling of functionally graded material cylindrical shells surrounded by a Paster-nak elastic foundation Theoretical formulations were presented based on a higher-order shear deformation shell theory Moreover, the post-buckling of FGM cylindrical shells surrounded by an elastic medium was presented by Shen [30,31] Sofiyev [32,33] studied the buckling of FGM shells on an elastic foundation The buckling of a heterogeneous orthotropic truncated conical shell under an axial load and surrounded by elastic media based on the finite deformation theory was investigated by Sofiyev [34] The governing equations of elastic buckling of heterogeneous orthotropic truncated conical shells using von Kármán nonlinearity were given Furthermore, Sofiyev [35] researched the nonlinear buckling of the FGM truncated conical shell sur-rounded by an elastic medium using the large deformation theory with von Kármán –Donnell type of kinematic nonlinearity

Stein and McElman [36] carried out the buckling problem of homogenous and isotropic toroidal shell segments Moreover, the initial post-buckling behavior of toroidal shell segments subject to several loading

conditions based on Koiter’s general theory was performed by Hutchinson [37] Parnell [38] gave a simple technique for the analysis of shells of revolution applied to toroidal shell segments

To the best of the authors’ knowledge, there has not been a study on the nonlinear torsional buckling of eccentrically stiffened FGM toroidal shell segments

In the present paper, the nonlinear torsional buckling and post-buckling of eccentrically stiffened FGM toroidal shell segments surrounded by an elastic medium are investigated Basing on the classical shell theory with nonlinear strain–displacement relation of large deflection, the Galerkin method is used for nonlinear buckling analysis of shells to give the expression of curves between deflection and torsional load The effects

of buckling modes, geometrical parameters, and volume fraction index on the nonlinear torsional buckling behavior of shells are investigated

2 Governing equations

2.1 Functionally graded material (FGM)

Suppose that the material composition of the shell varies smoothly along the thickness in such a way that the inner surface is ceramic rich and the outer surface is metal rich by a simple power law in terms of the volume fractions of the constituents

Denote Vmand Vc the volume fractions of metal and ceramic phases, respectively, which are related by

Vm+ Vc = 1 and Vcis expressed as Vm(z) =2z+h 2h k , where h is the thickness of the thin-walled structure,

k is the volume-fraction exponent (k ≥ 0); z is the thickness coordinate and varies from −h/2 to h/2; the

subscripts m and c refer to the metal and ceramic constituents, respectively According to the mentioned law, Young’s modulus reads:

E (z) = EmVm+ EmVm= Em+ (Em− Em)



2z + h 2h

k

Poisson’s ratioυ is assumed to be constant.

2.2 Constitutive relations and governing equations

Consider a functionally graded toroidal shell segment of thickness h and length L, which is formed by rotation

of a plane circular arc of radius R about an axis in the plane of the curve as shown in Fig.1 For the middle surface of a toroidal shell segment, from the figure:

r = a − R(1 − sin ϕ), where a is the equator radius and ϕ is the angle between the axis of revolution and the normal to the shell surface.

For a sufficiently shallow toroidal shell in the region of the equator of the torus, the angleϕ is approximately

equal toπ/2; thus, sin ϕ ≈ 1, cos ϕ ≈ 0, and r = a [36] The form of governing equation is simplified by putting:

dx1= R dϕ, dx2= a dθ.

The radius of arc R is positive with convex toroidal shell segment and negative with concave toroidal shell

segment

Suppose the FGM toroidal shell segment is reinforced by string and ring stiffeners In order to provide continuity within the shell and stiffeners and easier manufacture, homogeneous stiffeners can be used Because pure ceramic ones show brittleness, we used metal stiffeners and put them at the metal-rich side of the shell With the law indicated in (1), the outer surface is metal rich, so the external metal stiffeners are put at the outer side of the shell

The strains across the shell thickness at a distance z from the mid-surface are:

ε1= ε0

1− zχ1; ε2= ε0

2− zχ2; γ12= γ0

whereε0

1andε0

2are normal strains,γ0

12is the shear strain at the middle surface of the shell, andχ i j are the curvatures

Fig 1 Configuration of toroidal shell segments

According to the classical shell theory, the strains at the middle surface and curvatures are related to the

displacement components u , v, w in the x1, x2, z coordinate directions as [39]:

ε0

1= ∂u

∂x1 −w

R +1 2



∂w

∂x1

2

; ε0

2 = ∂v

∂x2 −w

a +1 2



∂w

∂x2

2

;

γ0

12= ∂u

∂x2 + ∂v

∂x1 + ∂w

∂x1

∂w

∂x2; χ1= ∂2w

∂x2 1

; χ2= ∂2w

∂x2 2

; χ12= ∂2w

From Eq (3), the strains must be satisfied in the deformation compatibility equation

∂2ε0 1

∂x2 2

+∂2ε20

∂x2 1

− ∂2γ120

∂x1∂x2 = − ∂2w

R ∂x2 2

− ∂2w

a ∂x2 1

+



∂2w

∂x1∂x2

2

−∂2w

∂x2 1

∂2w

∂x2 2

Hooke’s stress–strain relation is applied for the shell,

σ sh

1 = E (z)

1− ν2(ε1+ νε2),

σ sh

2 = E (z)

1− ν2(ε2+ νε1),

σ sh

2(1 + ν) γ12.

(5)

And for metal stiffeners

σ st

1 = Emε1,

σ st

Taking into account the contribution of stiffeners by the smeared stiffeners technique and omitting the twist

of stiffeners and integrating the stress–strain equations and their moments through the thickness of the shell,

we obtain the expressions for force and moment resultants of ES-FGM toroidal shell segment:

N1=



A11+ EmA1

s1



ε0

1+ A12ε0

2− (B11+ C1)χ1− B12χ2,

N2= A12ε0

1+



A22+ EmA2

s2



ε0

2− B12χ1− (B22+ C2)χ2, (7)

N12= A66γ0

12− 2B66χ12,

M1= (B11+ C1)ε0

1+ B12ε0

2−



D11+ EmI1

s1



χ1− D12χ2,

M2= B12ε0

1+ (B22+ C2)ε0

2− D12χ1−



D22+ EmI2

s2



M12= B66γ0

12− 2D66χ12

where A i j , B i j , D i j (i, j = 1, 2, 6) are extensional, coupling, and bending stiffnesses of the shell without

stiffeners

A11= A22= E1

1− ν2, A12= E1.ν

1− ν2, A66= E1

2(1 + ν) ,

B11= B22= E2

1− ν2, B12= E2.ν

1− ν2, B66= E2

2(1 + ν) ,

D11= D22= E3

1− ν2, D12= E3.ν

1− ν2, D66= E3

and



Em+ Em− Em

k+ 1



h , E2= (Em− Em)kh2

2(k + 1)(k + 2) ,

E3=



Em

12 + (Em− Em)

 1

k+ 3−

1

k+ 2+

1

4k+ 4



h3,

(10)

and

C1= ±EmA1z1

s1 , C2= ±EmA2z2

s2 ,

I1= d1h31

12 + A1z21, I2= d2h32

12 + A2z22.

In the above relations (7), (8), (10), and (11), Em is the elasticity modulus of the metal stiffener which is assumed to be identical for both types of stiffeners The spacings of the stringer and ring stiffeners are denoted

by s1and s2, respectively The quantities A1, A2are the cross section areas of the stiffeners, and I1, I2, z1, z2

are the second moments of cross section areas and eccentricities of the stiffeners with respect to the middle

surface of the shell, respectively The sign minus of C1and C2depends on external stiffeners

Remark Conversely, if the inner side of FGM shell is metal rich with existence of metal stiffeners, all calculated expressions can be used, but one must replace Ecand Emeach to other in Eq (10), and the plus sign is taken

in Eq (11)

The nonlinear equilibrium equations of a toroidal shell segment surrounded by an elastic foundation based

on the classical shell theory are given by [39]:

∂ N1

∂x1 +∂ N12

∂ N12

∂x1 +∂ N2

∂2M1

∂x2 1

+ 2∂2M12

∂x1∂x2 +∂2M2

∂x2 2

+ N1∂2w

∂x2 1

+ 2N12 ∂2w

+ N2∂2w

∂x2 2

+ N1

R + N2

a − K1w + K2



∂2w

∂x2 1

+∂2w

∂x2 2

where K1(N/m3) is the linear stiffness of the foundation and K2(N/m) is the shear modulus of the subgrade.

Considering the first two of Eqs (12), a stress function may be defined as:

N11=∂2F

∂x2 2

, N1

2 = ∂2F

∂x2 1

, N1

12= − ∂2F

The reverse relations are obtained from Eq (7)

ε0

1 = A∗22N1− A∗12N2+ B11∗χ1+ B12∗χ2,

ε0

2 = A∗11N2− A∗12N1+ B21∗χ1+ B22∗χ2, (14)

γ0

12= A∗66N12+ 2B66∗χ12,

where

A∗

11= 1



A11+ E0A1

s1



, A∗22=1



A22+ E0A2

s2



, A∗12= A12

, A∗66= 1

A66, =



A11+ E0A1

s1



.



A22+ E0A2

s2



− A2

12;

B∗

11= A∗

22(B11+ C1) − A∗

12B12, B∗

22= A∗

11(B22+ C2) − A∗

12B12,

B∗

12= A∗

22B12− A∗

12(B22+ C2), B∗

21= A∗

11.B12− A∗

12(B11+ C1), B∗

66= B66

A66.

Substituting Eq (14) in Eq (8) yields

M1= B11∗ N1+ B21∗ N2− D∗11χ1− D∗12χ2,

M2 = B∗

12N1+ B∗

22N2− D∗

21χ1− D∗

M12= B66∗ N12− 2D66∗χ12

where

D∗

11= D11+ E0I1

s1 − (B11+ C1)B11∗ − B12B∗

21,

D∗

22= D22+ E0I2

s2 − B12B∗

21− (B22+ C2)B22∗,

D∗

12= D12− (B11+ C1)B12∗ − B12B∗

22,

D∗

21= D12− B12B∗

11− (B22+ C2)B∗

21,

D∗

66= D66− B66B∗

66.

The substitution of Eq (14) in the compatibility Eqs (4) and (15) in Eq (12.3), taking into account expressions (3) and (13), yields a system of equations

A∗

11

∂4F

∂x4

1

+ (A∗

66− 2A∗

12) ∂4F

∂x2

1∂x2 2

+ A∗ 22

∂4F

∂x4 2

+ B∗ 21

∂4w

∂x4 1

+ (B∗

11+ B∗

22− 2B∗

66) ∂4w

∂x2

1∂x2 2

+

+ B12∗ ∂4w

∂x4

2

= −1

R

∂2w

∂x2 2

− 1

a

∂2w

∂x2 1

+



∂2w

∂x1∂x2

2

−∂2w

∂x2 1

∂2w

∂x2 2

B∗

21

∂4F

∂x4

1

+ (B11∗ + B22∗ − 2B66∗) ∂4F

∂x2

1∂x2 2

+ B12∗ ∂4F

∂x4 2

− D11∗ ∂4w

∂x4 1

− (D12∗ + D21∗ + 4D66∗ ) ∂4w

∂x2

1∂x2 2

− D22∗

∂4w

∂x4

2

+ 1

R

∂2F

∂x2 2

+1

a

∂2F

∂x2 1

+∂2F

∂x2 2

∂2w

∂x2 1

− 2∂x ∂2F

1∂x2

∂2w

∂x1∂x2 +∂2F

∂x2 1

∂2w

∂x2 2

− K1w + K2



∂2w

∂x2 1

+∂2w

∂x2 2

= 0.

(17)

3 Nonlinear torsional buckling analysis

The FGM toroidal shell segment is assumed to be simply supported at its edges x1 = 0 and x1 = L and

subjected to torsional load on the circular base of the shell

The edge is simply supported and freely movable (FM) in the axial direction The associated boundary conditions are:

w = 0, M1= 0, N1= 0, N12= τh at x1= 0; L. (18) With the consideration of boundary conditions (18), the deflection of the shell in this case can be expressed by [7]:

w = W0+ W1sinγ m x1sinβ n (x2− λx1) + W2sin2γ m x1, (19)

in which γ m = m π

L , β n = n

a , and m , n are the half wave numbers along x1-axis and wave numbers along

x2-axis, respectively The first term of w in Eq (19) represents the uniform deflection of points belonging

to two butt ends x1 = 0 and x1 = L, the second term—a linear buckling shape, and the third—a nonlinear

buckling shape

As can be seen, the simply supported boundary condition at x1= 0 and x1= L is fulfilled in the average

sense

Substituting Eq (19) in Eq (16) one obtains

A∗

11

∂4F

∂x4

1

+ (A∗66− 2A∗12) ∂4F

∂x2

1∂x2 2

+ A∗22

∂4F

∂x4 2

= H01cos 2γ m x1+ H02cos 2β n (x2− λx1) + H03 cosβ n



x2−



3γ m

β n + λ



x1



− cos β n



x2+



3γ m

β n − λ



x1



+ H04cosβ n



x2−



γ m

β n + λ



x1



+ H05cosβ n



x2+



γ m

β n − λ



x1



(20) where

H01=



2γ2

m



4B∗

21γ2

m− 1

a



W2+1

2W

2

1γ2

m β2

n ; H02= 1

2γ2

m β2

n W12; H03= 1

2γ2

m β2

n W1W2;

H04= 1

2W1 −B∗

21(γ2

m + β2

n λ2)2+ (2γ m β n λ)2+

 1

a − β2

n (B∗

11+ B∗

22− 2B∗

66)



(γ2

m + β2

n λ2) − B∗

12β4

n

+ 2γ m β n λ



−2B21∗(γ2

m + β2

n λ2) + 1

a − (B11∗ + B22∗ − 2B66∗)β2

n



−1

2γ2

m β2

n W1W2+ 1

2R β2

n W1;

H05= W1

1

2B

∗ 21

(γ2

m + β2

n λ2)2+ (2γ m β n λ)2 −1

2

 1

a − β2

n (B11∗ + B22∗ − 2B66∗ )



(γ2

m + β2

n λ2) + B12∗γ4

n

+ γ m β n λ



−2B21∗ (γ2

m + β2

n λ2) +1

a − (B11∗ + B22∗ − 2B66∗)β2

n

 +1

2γ2

m β2

n W1W2− 1

2R β2

n W1 (21)

The general solution of Eq (20) for a torsion-loaded shell is of the form

F = H1cos 2γ m x1+ H2cos 2β n (x2− λx1)

+ H3cosβ n



x2−



3γ m

β n + λ



x1



+ H4cosβ n



x2+



3γ m

β n − λ



x1



+ H5cosβ n



x2−

γ

m

β n + λ



x1



+ H6cosβ n



x2+

γ

m

β n − λ



x1



− τhx1x2 (22) whereτ is the torsional load intensity and the coefficients H i (i = 1 ÷ 8) are defined by:

H1= H01

16γ4

m A∗ 11

= M1W2+ M2W12;

16β4

n [A∗

11λ4+A∗

66− 2A∗ 12



λ2+ A∗

22] = M3W12;

β4

n



A∗ 11



3γ m

β n + λ4+A∗

66− 2A∗ 12

 3γ m

β n + λ2+ A∗

22

 = M4W1W2;

β4

n



A∗ 11

3γ

m

β n − λ4+A∗

66− 2A∗ 12

 3γm

β n − λ2+ A∗

22

 = M5W1W2;

β4

n



A∗ 11

γ

m

β n + λ4+A∗

66− 2A∗12 γ m

β n + λ2+ A∗22

 = M6W1+ M7W1W2;

β4

n



A∗ 11

γ

m

β − λ4+A∗

66− 2A∗12 γ m

β − λ2+ A∗22

 = M8W1+ M9W1W2 (23)

in which

M1= 4B21∗γ2

m−1

a

8γ2

m A∗

11

; M2= β n2

32γ2

m A∗ 11

32β2

n [A∗

11λ4+A∗

66− 2A∗ 12



λ2+ A∗

22];

β2

n



A∗

11



3γ m

β n + λ4+A∗

66− 2A∗ 12

 3γ m

β n + λ2+ A∗

22

;

β2

n



A∗

11

3γ

m

β n − λ4+A∗

66− 2A∗ 12

 3γm

β n − λ2+ A∗

22

;

M6=

1

2



−B∗

21(γ2

m + β2

n λ2)2+ (2γ m β n λ)2+1

a − β2

n (B∗

11+ B∗

22− 2B∗

66) (γ2

m + β2

n λ2) − B∗

12β4

n

+ 2γ m β n λ−2B21∗(γ2

m + β2

n λ2) + 1

a − (B11∗ + B22∗ − 2B66∗)β2

n + 1

2R β2

n

β4

n



A∗ 11

γ

m

β n + λ4+A∗

66− 2A∗ 12

 γ m

β n + λ2+ A∗

22

1

2γ2

m

β2

n



A∗

11

γ

m

β n + λ4+A∗

66− 2A∗ 12

 γ m

β n + λ2+ A∗

22

;

M8=



1

2B∗

21

(γ2

m + β2

n λ2)2+ (2γ m β n λ)2 − 1

2

1

a − β2

n (B∗

11+ B∗

22− 2B∗

66) (γ2

m + β2

n λ2) + B∗

12γ4

n

+ γ m β n λ−2B∗

21(γ2

m + β2

n λ2) +1

a − (B∗

11+ B∗

22− 2B∗

66)β2

n − 1

2R β2

n

β4

n



A∗ 11

γ

m

β n − λ4+A∗

66− 2A∗12 γ m

β n − λ2+ A∗22

m

β2

n



A∗

11

γ

m

β n − λ4+A∗

66− 2A∗ 12

 γ m

β n − λ2+ A∗

22

Equation (17) will be evaluated by the Galerkin method The procedure is performed in the following: Substituting Eqs (19) and (22) in the left side of Eq (17), then multiplying the obtained equation in turn with each shape function of Eq (19), and integrating in the ranges 0≤ x1≤ L; 0 ≤ x2≤ 2πa and after some

calculations lead to:

S1+ S2W2+ S3W12+ S4W22+ 2τβ2λh = 0, (25)

S5W2+ S6W12+ S7W12W2+ 2K1W0= 0 (26) where

S1=



B∗

21(γ m + β n λ)4+ B12∗ β4

n + β2

n (B11∗ + B22∗ − 2B66∗ )(γ m + β n λ)2−β n2

R − (γ m + β n λ)2

a



M6

−



B∗

21(γ m − β n λ)4+ B12∗β4

n + β2

n (B11∗ + B22∗ − 2B66∗)(γ m − β n λ)2−β n2

R − (γ m − β n λ)2

a



M8

−D11∗

2

(γ m + β n λ)4+ (γ m − β n λ)4

− (D∗12+ D21∗ + 4D∗66)β3

n γ m λ − D∗22β4

n − K1− K2



γ2

m + β2

n λ2

− K2β2

n



,



B∗

21(γ m + β n λ)4+ B12∗ β4

n + β2

n (B11∗ + B22∗ − 2B66∗ )(γ m + β n λ)2−β n2

R − (γ m + β n λ)2

a



M7

−



B∗

21(γ m − β n λ)4+ B12∗β4

n + β2

n (B11∗ + B22∗ − 2B66∗)(γ m − β n λ)2−β n2

R − (γ m − β n λ)2

a



M9

+ (M6− M8) γ2

m β2

n − 2M1γ2

m β2

n



,

S3= −2M3β2

n



γ2

m + β2

n λ2

− 2M3β4

n λ2+ M2γ2

m β2

n + M3β4

n λ2 ,

S4= γ2

m β2

n (M5+ M7− M4− M9) ,

S5=



16B∗

21γ4

m−4γ m2

a



M1+ 8γ4

m D∗

11+3K1

2 + 2K2γ2

m



,

S6=



16B∗

21γ4

m−4γ m2

a



M2+ M8β2

n (γ2

m + β2

n λ2− γ2

m β2

n λ2) − M6β2

n (γ2

m + β2

n λ2− γ2

m β2

n λ2)



, (27)

S7= γ2

m β2

n (M4+ M9− M5− M7)

Furthermore, the toroidal shell segments have to also satisfy the circumferential closed condition [7,15] as:

L



0

2πa



0

∂v

∂x2

d x1d x2=

L



0

2πa



0



ε0

2+w

a −1 2



∂w

∂x2

2

Using Eqs (13), (14), and (19), the integral becomes:

8W0+ 4W2− W2

1a β2

Substituting W0in Eqs (26)–(29), then substituting W12in Eq (26) into Eq (25) leads to an equation repre-senting theτ ∼ W2relation as

τ = −

⎛

⎝S1+ W2S2+ K1− S5

S6+ S7W2+ K1a β2

4

S3W2+ W2

2S4

⎞

2β2

Equation (30) expresses the post-bucklingτ ∼ W2curves of stiffened FGM toroidal shell segments When

W2→ 0, Eq (30) becomes

τ = − S1

2β2

Equation (31) is used to show upper critical loads in case of a linear buckling shape

From Eq (19), it can be seen that the maximal deflection of the shells

locates at x1= i L/(2m), x2= jπa/(2n) + iλL/(2m), where i and j are odd integer numbers.

Solving W1and W0from Eqs (25), (26), and (29) with respect to W2and then substituting them in Eq (32) leads to

Wmax=a β n2

8

⎛

⎝ K1W2− S5W2

S6+ S7W2+K1a β2

4

⎞

⎠ +



K1W2− S5W2

S6+ S7W2+ K1a β2

4

+W2

Combining Eq (30) with Eq (33), the post-buckling load-maximal deflection curves of stiffened FGM toroidal shell segments can be derived

The FGM toroidal shell segment is assumed to be simply supported at its edges x1 = and x1 = L and< /i>... iλL/(2m), where i and j are odd integer numbers.

Solving W1and W0from Eqs (25), (26), and (29) with respect to W2and then substituting...

+W2

Combining Eq (30) with Eq (33), the post -buckling load-maximal deflection curves of stiffened FGM toroidal shell segments can be derived